# Questions tagged [stochastic-processes]

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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### Weak convergence in Skorohod topology

Let $D([0,T];R^d)$ be the space of càdlàg functions endowed with the usual Skorohod topology. $X_t(\omega):=\omega(t)$ denotes the usual canonical process. Assume that a family of probability ...

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### Discrete time process with linear mixing and multiplicative noise

Consider a stochastic process $\vec{x}^t\in R^N$ in discrete time $t\in N$ which develops according to
$$\vec{x}^{t+1}_i=s_i^t \sum_j A_{ij}\vec{x}^t_j$$
where $A\in R^{N \times N}$ is some matrix ...

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### Wiener measure in the space of functions of two or more variables

Wiener measure in the space of continuous functions of two variables had been introduced by J. Yeh in the 1960 paper "Wiener Measure in a Space of Functions of Two Variables" (AMS free access)
(p. ...

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### Interchanging expectation and supremum when using control processes

This is a question I have from stochastic control. I know that in general $\underset{y\in \mathcal Y} \sup \mathbb E\big[f(X,y)\big] \le \mathbb E\big[\underset{y\in \mathcal Y} \sup f(X,y)\big]$. I ...

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### Show that the Markov chain of random tiling is irreducible

The Markov chain picks one hexagon consisting of all three types of lozenge uniformly at random and flips it. This is a finite-state Markov chain.
Show that this Markov chain is irreducible....

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### Have stick-breaking priors with non-iid atoms been considered, and if not, why not?

Roughly speaking, a stick-breaking prior is a random discrete probability measure $P$ on a measurable space $\mathcal X$ of the form
$$P=\sum_{j\ge1}w_j\delta_{\theta_j}$$
where $(w_j)_{j\ge1}$ is a ...

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### Derivation of a differential equation from a SDE

Suppose there is a non-homogeneous Markov process with state space $\mathbb{R}_{+}$
driven by this McKean-Vlasov-tipe SDE:
$$ dY_t = a \mathbb{E}[Y_t]\ dt - b\ Y_t\ dt - Y_t\ dN_{aY_t}$$
where $...

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### Markov with epsilon memory and Quantitative Strong Markov property

We have a process $\{X_{t}\}_{t\geq 0}$ ,with fixed parameter $\epsilon>0$, starting from zero that satisfies
The process is strictly monotone $X_{t+r}-X_{t}>0$ with moments existing $p\in(-\...

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### A question about positive operator pregenerator [closed]

Thank you for reading.
My question was raised up when I tried to prove an example in the book of Liggett(1985), which is in P13 Example 2.3(a).
Here is a link of the page:
https://books.google.com/...

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### Expected supremum of normalised random walk

Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$.
Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix.
Define $S^k=...

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### Which random walk can generate gamma distribution in the limit?

Symmetric random walk, its probability distribution is binomial coefficient, in the continuous limit, is Gaussian distribution:
$\displaystyle e^{- x^{2}}$
What kind of random walk, its probability ...

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### Expected minimum number

$X_i$ iid with $P(X_i=j)=p_j$, $j=1, \dots, m$.
$\sum_{j=1}^m p_j = 1$.
Define $N = \min\{n>0:X_n = X_0\}$, compute $E(N)$.
I have two solutions, but different answers:
Solution 1
$E(N) = E(N\...

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### About martingales induced by iterative processes

Suppose I have a discrete stochastic process $\{ X_i \}_{i=1,\ldots..}$ defined as, $X_{i+1} = X_i + \nabla f(X_i) + \xi_i$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and $\xi_i \sim {\cal N}(0,...

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### Distribution of markov chain with a stopping time

I have a Markov chain $X_0, X_1, ..., X_\tau$, where $X_0$ is sampled from the stationary distribution and $\tau$ is a stopping time. Is it true that for any fixed $k$, $X_k$, given that $k \leq \tau$,...

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### Comparing noisy truncated RV with noisy regular RV

For some reason, I'm having difficulties proving something that is intuitively simple.
Assuming I have two a random variable, $x$ and $x^{truncated}$, where $x^{truncated}$ is the truncated version of ...